2D
Table of Content
2D#
Point in 2d#
$ p = xvec{u} + yvec{v} + o $
$ p = begin{bmatrix} x & y & 1 end{bmatrix}begin{bmatrix} vec{u} \ vec{u} \ o end{bmatrix} $
Translation#
$ p = begin{bmatrix} x & y & 1 end{bmatrix} rightarrow begin{bmatrix} x+a & y+b & 1 end{bmatrix} $
$ begin{bmatrix} x & y & 1 end{bmatrix} begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ a & b & 1 end{bmatrix} = begin{bmatrix} x+a & y+b & 1 end{bmatrix} $
Scaleing#
$ begin{bmatrix} x & y & 1 end{bmatrix} rightarrow begin{bmatrix} ax & by & 1 end{bmatrix} $
$ begin{bmatrix} x & y & 1 end{bmatrix} begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & 1 \ end{bmatrix}= begin{bmatrix} ax & by & 1 end{bmatrix} $
TST#
- Translate to origin
- Scale
- Translate back
$ begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ -a & -b & 1 end{bmatrix} begin{bmatrix} Sx & 0 & 0 \ 0 & Sy & 0 \ 0 & 0 & 1 \ end{bmatrix} begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ -a & -b & 1 end{bmatrix} $